The starting point for the literature is the survey by Jeff Lagarias ten years ago:
math/0309224 The 3x+1 problem: An annotated bibliography (1963--1999) (sorted by author). Jeffrey C. Lagarias. math.NT (math.DS).
https://arxiv.org/abs/math/0309224
Other papers from an arxiv Front search for Collatz and 3x+1:
math/0608208 The 3x+1 Problem: An Annotated Bibliography, II (2000-2009). Jeffrey C. Lagarias. math.NT (math.DS) https://arxiv.org/abs/math/0608208
arXiv:0910.1944 Stochastic Models for the 3x+1 and 5x+1 Problems. Alex V. Kontorovich, Jeffrey C. Lagarias. in: The Ultimate Challenge: The 3x+1 problem, Amer. Math. Soc.: Providence 2010, pp. 131--188. math.NT. https://arxiv.org/abs/0910.1944
math/0509175 Benford's law for the $3x+1$ function. Jeffrey C. Lagarias, K. Soundararajan. J. London Math. Soc. 74 (2006), 289--303. math.NT (math.PR). https://arxiv.org/abs/math/0509175
math/0412003 Benford's Law, Values of L-functions and the 3x+1 Problem. Alex V. Kontorovich, Steven J. Miller. Acta Arithmetica 120 (2005), no. 3, 269-297. math.NT (math.PR). https://arxiv.org/abs/math/0412003
and
https://arxiv.org/abs/math/0411140
https://arxiv.org/abs/math/0205002
https://arxiv.org/abs/math/0201102
https://arxiv.org/abs/math/0204170
There are many un-serious papers on this subject at the same site, I selected the above from papers that I read or authors whose other papers I have seen.
There is also a celebrated theorem by John H Conway in which he shows that the natural generalization of the Collatz problem, where different linear functions are applied to $n$ in several arithmetic progressions (partitioning all integers), can simulate a computer. A consequence is that it is undecidable to determine if such an iteration loops when started from $1$.
I would not recommend continuing with Hardy and Wright. What you need is strength on the basics, not a tourist's view of a wide range of topics. I would focus on problem solving. Do lots of cool problems, taken from various sources, at a level that is reasonably challenging, but not way out of range, so as to develop your problem-solving and proof skills.
Best Answer
You may want to consider Collatz conjecture (aka the $3n+1$ conjecture).
In conjunction with Computer Science, you may give an overview of certain computational techniques and algorithms that have been used in order to analyze this conjecture.